#8712436

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,11)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(35,39)(36,42)(43,47)(44,48);(3,12)(4,13)(8,14)(9,16)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619632);